Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals.

A mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals.

On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexity.

Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, coordinational and configurational complexity, taking into account the individual multiplicities (combinatorial degrees of freedom) and arities (coordinational degrees of freedom) associated with each Wyckoff position. However, distinct Wyckoff sequences can yield the same total numbers of combinatorial and coordinational degrees of freedom.

On the combinatorics of crystal structures: number of Wyckoff sequences of given length.

A formula for the calculation of the number of Wyckoff sequences of a given length is presented, based on the combinatorics of multisets with finite multiplicities and a generating function approach, assuming a certain space-group type and taking into account the number of non-fixed and fixed Wyckoff positions, respectively. The formula is applied to the 44 distinguishable combinatorial types of the 230 space-group types. A comparison is made between the calculated frequencies of occurrence of Wyckoff sequences of given space-group type and length and the observed ones for actual crystal structures, as retrieved from the Pearson's Crystal Data Crystal Structure Database for Inorganic Compounds.

Chiral spiral cyclic twins.

A formula is presented for the generation of chiral m-fold multiply twinned two-dimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three non-degenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal long-range orientational order and symmetry.

Molybdenum Disorder in Hydrated Sedovite, Ideally U(MoO)·HO, a Microporous Nanocrystalline Mineral Characterized by Three-Dimensional Electron Diffraction, Density Functional Theory Computations, and Complexity Analysis.

Sedovite, U(MoO)·HO, is reported as being one of the earliest supergene minerals formed of the secondary zone. The difficulty of isolating enough pure material limits studies to techniques that can access the nanoscale combined with theoretical analyses. The crystal structure of sedovite has been solved and refined using the dynamical approach from three-dimensional electron diffraction data collected on natural nanocrystals found among iriginite.

The structures and phase transitions in 4-aminopyridinium tetraaquabis(sulfato)iron(III), (CHN)[Fe(HO)(SO)].

The organic-inorganic hybrid compound 4-aminopyridinium tetraaquabis(sulfato)iron(III), (CHN)[Fe(HO)(SO)] (4apFeS), was obtained by slow evaporation of the solvent at room temperature and characterized by single-crystal X-ray diffraction in the temperature range from 290 to 80 K. Differential scanning calorimetry revealed that the title compound undergoes a sequence of three reversible phase transitions, which has been verified by variable-temperature X-ray diffraction analysis during cooling-heating cycles over the temperature ranges 290-100-290 K. In the room-temperature phase (I), space group C2/c, oxygen atoms from the closest Fe-atom environment (octahedral) were disordered over two equivalent positions around a twofold axis.

On an extension of Krivovichev's complexity measures.

An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions.

A tenfold twin of the CrB structure type.

NiZr crystallized from an amorphous matrix or solidified from an undercooled melt exhibits a tenfold twinned microstructure, which is explained by an ideal twin model utilizing special geometric properties of the CrB structure type. The model is unique in several ways: (i) it contains no adjustable parameters other than a scaling factor accounting for the smallest interatomic distance; (ii) it features an irrational shift in the translational part of the twin operation; and (iii) it has many traits commonly observed for quasicrystals, connected to the occurrence of decagonal long-range orientational order, making NiZr the first experimental example of the recently introduced concept of {\bb Z}-module twinning. It is shown how these remarkable properties of the tenfold twin's structure model are related to one another and founded in number theory as well as in the mathematical theory of aperiodic order.

Quasicrystal nucleation and [Formula: see text] module twin growth in an intermetallic glass-forming system.

While quasicrystals possess long-range orientational order they lack translation periodicity. Considerable advancements in the elucidation of their structures and formative principles contrast with comparatively uncharted interrelations, as studies bridging the spatial scales from atoms to the macroscale are scarce. Here, we report on the homogeneous nucleation of a single quasicrystalline seed from the undercooled melt of glass-forming NiZr and its continuous growth into a tenfold twinned dendritic microstructure.

Structural chemistry and number theory amalgamized: crystal structure of Na11Hg52.

The recently elucidated crystal structure of the technologically important amalgam Na11Hg52 is described by means of a method employing some fundamental concept of number theory, namely modular arithmetical (congruence) relations observed between a slightly idealized set of atomic coordinates. In combination with well known ideas from group theory, regarding lattice-sublattice transformations, these allow for a deeper mutual understanding of both and provide the structural chemist with a slightly different kind of spectacles, thus enabling a distinct viw on complex crystal structures in general.

Diaphony, a measure of uniform distribution, and the Patterson function.

This paper reviews the number-theoretic concept of diaphony, a measure of uniform distribution for number sequences and point sets based on a Fourier theory approach, and its relation to crystallographic concepts like the largest interplanar spacing of a lattice, the structure-factor equation and the Patterson function.

Nucleation and crystal growth in a suspension of charged colloidal silica spheres with bi-modal size distribution studied by time-resolved ultra-small-angle X-ray scattering.

A suspension of charged colloidal silica spheres exhibiting a bi-modal size distribution of particles, thereby mimicking a binary mixture, was studied using time-resolved ultra-small-angle synchrotron X-ray scattering (USAXS). The sample, consisting of particles of diameters d(A) = (104.7 ± 9.

Octagonal symmetry in low-discrepancy β-manganese.

A low-discrepancy cubic variant of β-Mn is presented exhibiting local octagonal symmetry upon projection along any of the three mutually perpendicular 〈100〉 axes. Ideal structural parameters are derived to be x(8c) = (2-\sqrt{2})\big/16 and y(12d) = 1\big/(4 \sqrt{2}) for the P4132 enantiomorph. A comparison of the actual and ideal structure models of β-Mn is made in terms of the newly devised concept of geometrical discrepancy maps.

Quantitative crystal structure descriptors from multiplicative congruential generators.

Special types of number-theoretic relations, termed multiplicative congruential generators (MCGs), exhibit an intrinsic sublattice structure. This has considerable implications within the crystallographic realm, namely for the coordinate description of crystal structures for which MCGs allow for a concise way of encoding the numerical structural information. Thus, a conceptual framework is established, with some focus on layered superstructures, which proposes the use of MCGs as a tool for the quantitative description of crystal structures.

Multiplicative congruential generators, their lattice structure, its relation to lattice-sublattice transformations and applications in crystallography.

An analysis of certain types of multiplicative congruential generators--otherwise known for their application to the sequential generation of pseudo-random numbers--reveals their relation to the coordinate description of lattice points in two-dimensional primitive sublattices. Taking the index of the lattice-sublattice transformation as the modulus of the multiplicative congruential generator, there are special choices for its multiplier which induce a symmetry-preserving permutation of lattice-point coordinates. From an analysis of similar sublattices with hexagonal and square symmetry it is conjectured that the cycle structure of the permutation has its crystallographic counterpart in the description of crystallographic orbits.

Structure-composition relations for the partly disordered Hume-Rothery phase Ir7+7deltaZn97-11delta (0.31< or =delta< or =0.58).

The crystal structure, its variation within the homogeneity range and some physical properties of the new zinc-rich, partly disordered phase Ir7+7deltaZn97-11delta (0.31< or =delta< or =0.58) are reported.